Finite Group Databasebeta

← C31×#23546 #21480305

Subgroups — C31×#23546 #21480305

Order 7936.

1674 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×AGL(1,ℤ/8)C211yesmaximal, characteristic
2C2×C31×#423C211yesmaximal, characteristic, complemented
3C2×C31×#417C211yesmaximal, characteristic, complemented
4C2×C31×#418C211yesmaximal, characteristic, complemented
5C31×#2297C211yesmaximal, characteristic, complemented
6C2×C31×#390C211yesmaximal, characteristic, complemented
7C31×#2055C214yesmaximal, complemented
8C31×#2054C214yesmaximal, complemented
9C31×#1736C211yesmaximal, characteristic, complemented
10C22×#41×C31C2211yescharacteristic
11C23×D4×C31C2211yescharacteristic
12C2×C31×AGL(1,ℤ/8)—24no—
13C2×C31×AGL(1,ℤ/8)—24no—
14C2×C31×AGL(1,ℤ/8)—24no—
15C22×SD16×C31C2212yes—
16C22×D8×C31C2212yes—
17C22×M16×C31C2211yescharacteristic
18C22×M16×C31C2211yescharacteristic
19C2×C31×#117C2211yescharacteristic
20C22×C4⋊C4×C31C2211yescharacteristic
21C22×C22⋊C4×C31—22no—
22C22×C42×C31C2211yescharacteristic
23C31×#423—24no—
24C2×C31×C4⋊C8C2214yescomplemented
25C31×#418—24no—
26C31×#417—24no—
27C2×C31×#103C2212yescomplemented
28C2×C31×#102C2214yes—
29C2×C31×#102C2212yescomplemented
30C31×#391C2212yescomplemented
31C31×#390—21no—
32C31×#390—21no—
33C31×#390C2214yes—
34C31×#390C2212yescomplemented
35C31×#386—22no—
36C4×C22⋊C4×C31—21no—
37C4×C22⋊C4×C31—21no—
38C31×#340C2214yescomplemented
39C25×C31—21no—
40C2×#41×C31—22no—
41C2×#41×C31—22no—
42C2×#41×C31—22no—
43C2×#41×C31—22no—
44C2×#41×C31—22no—
45C2×#41×C31D414yes—
46C22×Q8×C31C2311yescharacteristic
47C22×D4×C31—21no—
48C22×D4×C31—22no—
49C22×D4×C31—22no—
50C22×D4×C31—21no—
51C22×D4×C31—22no—
52C22×D4×C31—22no—
53C22×D4×C31—22no—
54C22×D4×C31D414yes—
55C22×D4×C31C2312yes—
56C22×D4×C31C2311yescharacteristic
57C23×C4×C31—21no—
58C23×C4×C31—21no—
59C23×C4×C31D412yes—
60C23×C4×C31C2311yescharacteristic
61C31×AGL(1,ℤ/8)—232no—
62C2×SD16×C31—24no—
63C2×SD16×C31—24no—
64C2×SD16×C31—24no—
65C2×D8×C31—24no—
66C2×D8×C31—24no—
67C2×D8×C31—24no—
68C2×M16×C31—22no—
69C2×M16×C31—22no—
70C2×M16×C31—22no—
71C2×M16×C31—22no—
72C2×M16×C31—22no—
73C2×M16×C31D414yes—
74C22×C8×C31C2312yes—
75C22×C8×C31C2312yes—
76C31×#117—22no—
77C31×#117—22no—
78C2×C4⋊C4×C31—22no—
79C2×C4⋊C4×C31—22no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31C2312yes—
82C2×C4⋊C4×C31C2312yes—
83C2×C22⋊C4×C31—21no—
84C2×C22⋊C4×C31—24no—
85C2×C22⋊C4×C31—24no—
86C2×C22⋊C4×C31—22no—
87C2×C22⋊C4×C31—22no—
88C2×C22⋊C4×C31—48no—
89C2×C42×C31—22no—
90C2×C42×C31—22no—
91C2×C42×C31—21no—
92C2×C42×C31D411yescharacteristic
93C2×C42×C31D411yescharacteristic
94C2×C42×C31C2312yes—
95C31×C4⋊C8—28no—
96C31×#103—24no—
97C31×#102—28no—
98C31×#102—24no—
99C31×#95—22no—
100C31×#95—22no—
101C24×C31—24no—
102C24×C31—21no—
103C24×C31—22no—
104C24×C31—44no—
105C24×C31—44no—
106C24×C31—42no—
107C24×C31—42no—
108C24×C31C2×D411yescharacteristic
109#41×C31—28no—
110#41×C31—28no—
111#41×C31—44no—
112#41×C31—44no—
113C2×Q8×C31—21no—
114C2×Q8×C31—21no—
115C2×Q8×C31—21no—
116C2×Q8×C31—21no—
117C2×Q8×C31—21no—
118C2×Q8×C31C2×D412yes—
119C2×D4×C31—24no—
120C2×D4×C31—28no—
121C2×D4×C31—24no—
122C2×D4×C31—28no—
123C2×D4×C31—21no—
124C2×D4×C31—21no—
125C2×D4×C31—22no—
126C2×D4×C31—22no—
127C2×D4×C31—22no—
128C2×D4×C31—22no—
129C2×D4×C31—21no—
130C2×D4×C31—21no—
131C2×D4×C31—22no—
132C2×D4×C31—21no—
133C2×D4×C31—42no—
134C2×D4×C31—42no—
135C2×D4×C31—42no—
136C2×D4×C31—42no—
137C2×D4×C31—44no—
138C2×D4×C31—44no—
139C2×D4×C31C2×D412yes—
140C2×D4×C31C2×D414yes—
141C22×C4×C31—24no—
142C22×C4×C31—21no—
143C22×C4×C31—24no—
144C22×C4×C31—24no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—42no—
148C22×C4×C31—42no—
149C22×C4×C31—42no—
150C22×C4×C31—44no—
151C22×C4×C31—42no—
152C22×C4×C31C2×D412yes—
153C22×C4×C31C2×D412yes—
154C22×C4×C31C2×D412yes—
155C22×C4×C31C2×D412yes—
156C22×C4×C31C2×D412yes—
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31#4112yes—
159C22×C4×C31#4112yes—
160C22×C4×C31#4112yes—
161C22×C4×C31C2411yescharacteristic
162SD16×C31—216no—
163D8×C31—216no—
164M16×C31—28no—
165M16×C31—44no—
166C2×C8×C31—22no—
167C2×C8×C31—22no—
168C2×C8×C31—22no—
169C2×C8×C31—22no—
170C2×C8×C31—22no—
171C2×C8×C31C2×D414yes—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—22no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—42no—
177C22⋊C4×C31—44no—
178C22⋊C4×C31—48no—
179C22⋊C4×C31—42no—
180C22⋊C4×C31—44no—
181C22⋊C4×C31—48no—
182C42×C31—22no—
183C42×C31—22no—
184C42×C31—22no—
185C42×C31—22no—
186C42×C31—42no—
187#23546C3111yesmaximal, characteristic, direct product factor, Sylow
188C23×C31—22no—
189C23×C31—22no—
190C23×C31—22no—
191C23×C31—24no—
192C23×C31—416no—
193C23×C31—41no—
194C23×C31—41no—
195C23×C31—42no—
196C23×C31—42no—
197C23×C31—44no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—44no—
201C23×C31—48no—
202C23×C31—84no—
203C23×C31—84no—
204C23×C31—88no—
205C23×C31#12112yes—
206C23×C31#12112yes—
207C23×C31#12112yes—
208C23×C31C22×D411yescharacteristic
209Q8×C31—22no—
210Q8×C31—22no—
211Q8×C31—41no—
212Q8×C31—41no—
213D4×C31—22no—
214D4×C31—24no—
215D4×C31—24no—
216D4×C31—22no—
217D4×C31—48no—
218D4×C31—42no—
219D4×C31—41no—
220D4×C31—42no—
221D4×C31—41no—
222D4×C31—48no—
223C2×C4×C31—22no—
224C2×C4×C31—22no—
225C2×C4×C31—22no—
226C2×C4×C31—22no—
227C2×C4×C31—22no—
228C2×C4×C31—22no—
229C2×C4×C31—48no—
230C2×C4×C31—41no—
231C2×C4×C31—41no—
232C2×C4×C31—48no—
233C2×C4×C31—42no—
234C2×C4×C31—42no—
235C2×C4×C31—44no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—84no—
239C2×C4×C31#12014yes—
240C2×C4×C31#12112yes—
241C2×C4×C31#12112yes—
242C2×C4×C31#12112yes—
243C2×C4×C31#12314yes—
244C2×C4×C31#12314yes—
245C2×C4×C31#12414yes—
246C2×C4×C31AGL(1,ℤ/8)14yescomplemented
247C2×C4×C31C22×D411yescharacteristic
248C2×C4×C31C22×D411yescharacteristic
249C2×C4×C31C22×D411yescharacteristic
250C2×C4×C31C2×#4111yescharacteristic
251C2×C4×C31C2×#4111yescharacteristic
252C2×C4×C31C2×#4111yescharacteristic
253C8×C31—24no—
254C8×C31—42no—
255C22×AGL(1,ℤ/8)C2×C3111yescharacteristic
256C2×#423C2×C3111yescharacteristic, complemented
257C2×#417C2×C3111yescharacteristic, complemented
258C2×#418C2×C3111yescharacteristic, complemented
259#2297C2×C3111yescharacteristic, complemented
260C2×#390C2×C3111yescharacteristic, complemented
261#2055C2×C3114yescomplemented
262#2054C2×C3114yescomplemented
263#1736C2×C3111yescharacteristic, complemented
264C22×C31—24no—
265C22×C31—22no—
266C22×C31—22no—
267C22×C31—24no—
268C22×C31—42no—
269C22×C31—44no—
270C22×C31—48no—
271C22×C31—42no—
272C22×C31—42no—
273C22×C31—816no—
274C22×C31—82no—
275C22×C31—82no—
276C22×C31—82no—
277C22×C31—82no—
278C22×C31—84no—
279C22×C31#39414yes—
280C22×C31C2×#12111yescharacteristic
281C22×C31C2×#12111yescharacteristic
282C22×C31C2×#12111yescharacteristic
283C22×C31C2×AGL(1,ℤ/8)12yes—
284C22×C31#57512yes—
285C4×C31—24no—
286C4×C31—42no—
287C4×C31—42no—
288C4×C31—82no—
289C4×C31—82no—
290C4×C31#39414yes—
291C4×C31C2×#12011yescharacteristic
292C4×C31C2×#12311yescharacteristic
293C4×C31C2×#12311yescharacteristic
294C4×C31C2×#12411yescharacteristic
295C22×#41C22×C3111yescharacteristic
296C23×D4C22×C3111yescharacteristic
297C2×AGL(1,ℤ/8)—24no—
298C2×AGL(1,ℤ/8)—24no—
299C2×AGL(1,ℤ/8)—24no—
300C22×SD16C22×C3112yes—
301C22×D8C22×C3112yes—
302C22×M16C22×C3111yescharacteristic
303C22×M16C22×C3111yescharacteristic
304C2×#117C22×C3111yescharacteristic
305C22×C4⋊C4C22×C3111yescharacteristic
306C22×C22⋊C4—22no—
307C22×C42C22×C3111yescharacteristic
308#423—24no—
309C2×C4⋊C8C22×C3114yescomplemented
310#418—24no—
311#417—24no—
312C2×#103C22×C3112yescomplemented
313C2×#102C22×C3114yes—
314C2×#102C22×C3112yescomplemented
315#391C22×C3112yescomplemented
316#390—21no—
317#390—21no—
318#390C22×C3114yes—
319#390C22×C3112yescomplemented
320#386—22no—
321C4×C22⋊C4—21no—
322C4×C22⋊C4—21no—
323#340C22×C3114yescomplemented
324C2×C31—24no—
325C2×C31—82no—
326C2×C31—84no—
327C2×C31C2×#39411yescharacteristic
328C2×C31#303612yes—
329C2×C31#303811yescharacteristic
330C2×C31#304011yescharacteristic
331C2×C31#305512yes—
332C25—21no—
333C2×#41—22no—
334C2×#41—22no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41D4×C3114yes—
339C22×Q8C23×C3111yescharacteristic
340C22×D4—21no—
341C22×D4—22no—
342C22×D4—22no—
343C22×D4—21no—
344C22×D4—22no—
345C22×D4—22no—
346C22×D4—22no—
347C22×D4D4×C3114yes—
348C22×D4C23×C3111yescharacteristic
349C22×D4C23×C3112yes—
350C23×C4—21no—
351C23×C4—21no—
352C23×C4D4×C3112yes—
353C23×C4C23×C3111yescharacteristic
354AGL(1,ℤ/8)—232no—
355C2×SD16—24no—
356C2×SD16—24no—
357C2×SD16—24no—
358C2×D8—24no—
359C2×D8—24no—
360C2×D8—24no—
361C2×M16—22no—
362C2×M16—22no—
363C2×M16—22no—
364C2×M16—22no—
365C2×M16—22no—
366C2×M16D4×C3114yes—
367C22×C8C23×C3112yes—
368C22×C8C23×C3112yes—
369#117—22no—
370#117—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4C23×C3112yes—
375C2×C4⋊C4C23×C3112yes—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—24no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—48no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—21no—
385C2×C42D4×C3111yescharacteristic
386C2×C42D4×C3111yescharacteristic
387C2×C42C23×C3112yes—
388C4⋊C8—28no—
389#103—24no—
390#102—28no—
391#102—24no—
392#95—22no—
393#95—22no—
394C31#2354611yescharacteristic, direct product factor, Sylow
395C24—24no—
396C24—21no—
397C24—22no—
398C24—44no—
399C24—44no—
400C24—42no—
401C24—42no—
402C24C2×D4×C3111yescharacteristic
403#41—28no—
404#41—28no—
405#41—44no—
406#41—44no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8C2×D4×C3112yes—
413C2×D4—24no—
414C2×D4—28no—
415C2×D4—24no—
416C2×D4—28no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4C2×D4×C3112yes—
434C2×D4C2×D4×C3114yes—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4C2×D4×C3112yes—
447C22×C4C2×D4×C3112yes—
448C22×C4C2×D4×C3112yes—
449C22×C4C2×D4×C3112yes—
450C22×C4C2×D4×C3112yes—
451C22×C4C2×D4×C3111yescharacteristic
452C22×C4#41×C3112yes—
453C22×C4#41×C3112yes—
454C22×C4#41×C3112yes—
455C22×C4C24×C3111yescharacteristic
456SD16—216no—
457D8—216no—
458M16—28no—
459M16—44no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8C2×D4×C3114yes—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—42no—
471C22⋊C4—44no—
472C22⋊C4—48no—
473C22⋊C4—42no—
474C22⋊C4—44no—
475C22⋊C4—48no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—42no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—416no—
486C23—41no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—48no—
495C23—84no—
496C23—84no—
497C23—88no—
498C23C31×#12112yes—
499C23C31×#12112yes—
500C23C31×#12112yes—
501C23C22×D4×C3111yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—41no—
505Q8—41no—
506D4—22no—
507D4—24no—
508D4—24no—
509D4—22no—
510D4—48no—
511D4—42no—
512D4—41no—
513D4—42no—
514D4—41no—
515D4—48no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—48no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—48no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—84no—
532C2×C4C31×#12014yes—
533C2×C4C31×#12112yes—
534C2×C4C31×#12112yes—
535C2×C4C31×#12112yes—
536C2×C4C31×#12314yes—
537C2×C4C31×#12314yes—
538C2×C4C31×#12414yes—
539C2×C4C31×AGL(1,ℤ/8)14yescomplemented
540C2×C4C22×D4×C3111yescharacteristic
541C2×C4C22×D4×C3111yescharacteristic
542C2×C4C22×D4×C3111yescharacteristic
543C2×C4C2×#41×C3111yescharacteristic
544C2×C4C2×#41×C3111yescharacteristic
545C2×C4C2×#41×C3111yescharacteristic
546C8—24no—
547C8—42no—
548C22—24no—
549C22—22no—
550C22—22no—
551C22—24no—
552C22—42no—
553C22—44no—
554C22—48no—
555C22—42no—
556C22—42no—
557C22—816no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—84no—
562C22—82no—
563C22C31×#39414yes—
564C22C2×C31×#12111yescharacteristic
565C22C2×C31×#12111yescharacteristic
566C22C2×C31×#12111yescharacteristic
567C22C2×C31×AGL(1,ℤ/8)12yes—
568C22C31×#57512yes—
569C4—24no—
570C4—42no—
571C4—42no—
572C4—82no—
573C4—82no—
574C4C31×#39414yes—
575C4C2×C31×#12011yescharacteristic
576C4C2×C31×#12311yescharacteristic
577C4C2×C31×#12311yescharacteristic
578C4C2×C31×#12411yescharacteristic
579C2—24no—
580C2—84no—
581C2—82no—
582C2C2×C31×#39411yescharacteristic
583C2C31×#303612yes—
584C2C31×#303811yescharacteristic
585C2C31×#304011yescharacteristic
586C2C31×#305512yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×C31×AGL(1,ℤ/8)C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C31×#423C2semidirect
3. C2×C31×#417C2semidirect
4. C2×C31×#418C2semidirect
5. C31×#2297C2semidirect
6. C2×C31×#390C2semidirect
7. C31×#2055C2semidirect
8. C31×#2054C2semidirect
9. C31×#1736C2semidirect
24. C2×C31×C4⋊C8C22semidirect
27. C2×C31×#103C22semidirect
29. C2×C31×#102C22semidirect
30. C31×#391C22semidirect
34. C31×#390C22semidirect
38. C31×#340C22semidirect
187. #23546C31direct (N × H)
246. C2×C4×C31AGL(1,ℤ/8)semidirect
256. C2×#423C2×C31semidirect
257. C2×#417C2×C31semidirect
258. C2×#418C2×C31semidirect
259. #2297C2×C31semidirect
260. C2×#390C2×C31semidirect
261. #2055C2×C31semidirect
262. #2054C2×C31semidirect
263. #1736C2×C31semidirect
309. C2×C4⋊C8C22×C31semidirect
312. C2×#103C22×C31semidirect
314. C2×#102C22×C31semidirect
315. #391C22×C31semidirect
319. #390C22×C31semidirect
323. #340C22×C31semidirect
394. C31#23546direct (N × H)
539. C2×C4C31×AGL(1,ℤ/8)semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C31×AGL(1,ℤ/8)2
C2×C31×#4232
C2×C31×#4172
C2×C31×#4182
C31×#22972
C2×C31×#3902
C31×#20552
C31×#20542
C31×#17362
#2354631